The speed of a motor increases from 1200 rpm to 1800 rpm in 20 s . How many revolutions does it make during these second ?

Options

A. 400
B. 600
C. 500
D. 700

Explanation

To solve the problem step by step, we will calculate the number of revolutions made by the motor as its speed increases from 1200 rpm to 1800 rpm over a period of 20 seconds.

### Step 1: Convert RPM to RPS
First, we need to convert the initial and final speeds from revolutions per minute (rpm) to revolutions per second (rps).

- Initial speed (ω_initial) = 1200 rpm
- Final speed (ω_final) = 1800 rpm

To convert rpm to rps, we use the conversion factor:
\[ \text{rps} = \frac{\text{rpm}}{60} \]

Calculating:
- \( \omega_{\text{initial}} = \frac{1200}{60} = 20 \, \text{rps} \)
- \( \omega_{\text{final}} = \frac{1800}{60} = 30 \, \text{rps} \)

### Step 2: Calculate Angular Acceleration (α)
Next, we need to find the angular acceleration (α) using the formula:
\[ \omega_{\text{final}} = \omega_{\text{initial}} + \alpha t \]

Rearranging the formula to solve for α:
\[ \alpha = \frac{\omega_{\text{final}} - \omega_{\text{initial}}}{t} \]

Substituting the values:
- \( t = 20 \, \text{s} \)

Calculating:
\[ \alpha = \frac{30 \, \text{rps} - 20 \, \text{rps}}{20 \, \text{s}} = \frac{10 \, \text{rps}}{20 \, \text{s}} = 0.5 \, \text{rps}^2 \]

### Step 3: Calculate the Total Revolutions (θ)
Now we can use the formula for angular displacement (θ) during uniformly accelerated motion:
\[ \theta = \omega_{\text{initial}} t + \frac{1}{2} \alpha t^2 \]

Substituting the known values:
- \( \theta = 20 \, \text{rps} \times 20 \, \text{s} + \frac{1}{2} \times 0.5 \, \text{rps}^2 \times (20 \, \text{s})^2 \)

Calculating:
1. First term: \( 20 \, \text{rps} \times 20 \, \text{s} = 400 \, \text{revolutions} \)
2. Second term:
\[ \frac{1}{2} \times 0.5 \, \text{rps}^2 \times 400 \, \text{s}^2 = 0.25 \times 400 = 100 \, \text{revolutions} \]

Adding both terms:
\[ \theta = 400 + 100 = 500 \, \text{revolutions} \]

### Final Answer
The motor makes **500 revolutions** during the 20 seconds.

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